$\begingroup$So I thought about it and if we consider the case when $|z|=3/4$, then let $ (z)=-e^z$ then we have $|g(z)+f(z)|=|z^3|=(3/4)^3<|f(z)|=e^{3/4},$ and therefore $f(z)$ has same number of roots as $g(z)$ and since $f(z)$ has no roots in $\mathbb{C}, g(z)$ also has no roots in the disk $|z|=3/4.$ Is this correct?$\endgroup$
– Aurora BorealisApr 24 '18 at 3:26